53 research outputs found
Global cycle properties in graphs with large minimum clustering coefficient
The clustering coefficient of a vertex in a graph is the proportion of
neighbours of the vertex that are adjacent. The minimum clustering coefficient
of a graph is the smallest clustering coefficient taken over all vertices. A
complete structural characterization of those locally connected graphs, with
minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully
cycle extendable is given in terms of strongly induced subgraphs with given
attachment sets. Moreover, it is shown that all locally connected graphs with
minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly
pancyclic, thereby proving Ryjacek's conjecture for this class of locally
connected graphs.Comment: 16 pages, two figure
The metric dimension of Cayley digraphs
AbstractA vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay(Ξ:Ξ), where Ξ is the group Zn1βZn2ββ―βZnm and Ξ is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0,1),(1,0)}:ZnβZm) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group Dn of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay(Ξ:Dn), where Ξ is a minimum set of generators for Dn, are established
Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs
Given a finite set , a convexity , is a collection of subsets
of that contains both the empty set and the set and is closed under
intersections. The elements of are called convex sets. The
digital convexity, originally proposed as a tool for processing digital images,
is defined as follows: a subset is digitally convex if, for
every , we have implies . The number of
cyclic binary strings with blocks of length at least is expressed as a
linear recurrence relation for . A bijection is established between
these cyclic binary strings and the digitally convex sets of the
power of a cycle. A closed formula for the number of digitally convex sets of
the Cartesian product of two complete graphs is derived. A bijection is
established between the digitally convex sets of the Cartesian product of two
paths, , and certain types of binary arrays.Comment: 16 pages, 3 figures, 1 tabl
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